Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No: 299.02083
Autor: Erdös, Paul; Hajnal, András; Shelah, Saharon
Title: On some general properties of chromatic numbers. (In English)
Source: Topics in Topol., Colloqu. Keszthely 1972, Colloquia Math. Soc. Janos Bolyai 8, 243-255 (1974).
Review: [For the entire collection see Zbl 278.00018.]
The starting point is the following Taylor problem [W. Taylor, Fundamenta Math. 71, 103-112 (1971; Zbl 238.02044); p. 106, problem 1.14]: What is the minimal cardinal number \lambda such that for every graph G with chromatic number \chi G \geq \lambda and every cardinal \sigma \geq \lambda there exists a graph G' such that \chi (G') \geq \sigma and that G,G' have the same finite graphs? According to Taylor, \lambda \geq \omega1, conjecturing \lambda = \omega1. The authors formulate 7 other problems, prove 3 theorems and several lemmas. E.g., if \chi (G) > \omega, then for some n < \omega the graph G contains odd circuits of length 2j+1 for every n < j < \omega (theorem 3). For any ordered set (R, \prec) and any i < \omega the authors define two sorts of graphs G0(R,i) for i > 1 and G1(R,i,t) for i > 2, 1 \leq t < i-1 as \prec-increasing i-sequences of points of R such that

for |R| \geq \omega; the graphs S0,S1 are called ``edge graphs'' and Specker graphs respectively. Notations: For any cardinality \tau \geq \omega let B(\tau) be the system of all subgraphs of cardinality < \tau of some complete graph with \tau vertices; put A(\tau) = P(B(\tau)). If G is a given graph let

For S in A(\tau) let G(S, \tau) be the class of graphs G satisfying \psi (G, \tau) \subset S in A(\tau); S is said \tau-unbounded if for every cardinal \lambda there is some G in G(S, \tau) satisfying \chi (G) > \lambda. For a given operation F on cardinals satisfying Fx \geq x^+, the authors say that S in A(\omega) is \omega-unbounded with the restriction F, if for every \sigma there is some \lambda \geq \sigma and a graph G such that \psi (G, \omega) \subset S, \chi (G) > \lambda and |G| \leq F(\lambda). In particular, S is \omega-unbounded with restriction \xi if S is so with the restriction F\xi where